# How do you write the partial fraction decomposition of the rational expression 3 / (x^2 - 3x)?

Jan 18, 2016

$- \frac{1}{x} + \frac{1}{x - 3}$

#### Explanation:

To write the partial fraction decomposition, first factorize the denominator
$f \left(x\right) = \frac{3}{{x}^{2} - 3 x} = \frac{3}{x \left(x - 3\right)}$
This can then be written as $\frac{A}{x} + \frac{B}{x - 3}$
Reformulating this over the common denominator gives
$\frac{A \left(x - 3\right) + B x}{x \left(x - 3\right)}$
$= \frac{\left(A + B\right) x - 3 A}{x \left(x - 3\right)}$
Therefore $\left(A + B\right) = 0$ and$- 3 A = 3$
$\therefore A = - 1$ and $B = 1$

The original expression can be written as
$- \frac{1}{x} + \frac{1}{x - 3}$